If the ball is extremely close to your own end zone and your offensive line false starts (or commits any infraction, really) and the ball cannot be moved any closer to the goal line, what happens? Loss of down? Can the offense continue to commit penalties and it stays first down?
This is actually an interesting question. There is a clause in the rules, 12.3.2 which covers repeated DEFENSIVE half distance penalties.
The defense shall not commit successive or repeated fouls to prevent a score.
Penalty: For successive or repeated fouls to prevent a score: If the violation is repeated after a warning, the score involved is awarded to the offensive team.
There doesn't appear to be an equivalent rule for the offense, although rule 12.3.4 which covers "palpably unfair acts" gives the Referee a large amount of leeway in covering unusual situations.
For a palpably unfair act: Offender may be disqualified. The Referee, after consulting the officiating crew, enforces any such distance penalty as they consider equitable and irrespective of any other specified code penalty. The Referee may award a score. See 19-1-3.
On that wording, it would be hard to argue against enforcing an equivalent to the defense penalty - so that if Team A has the ball, 1st and 10 on their own 1 yard line, and they commit three successive false starts (so they are now on the 1/8 yard line) the Referee should give a warning and, if they commit another penalty, give a safety to the defense.
Now, normally after a safety, the offense gets the chance to take a free kick from the 20 yard line - I would not be surprised if the Referee also enforces a half distance to the goal penalty and forces them to take the kick from the 10 yard line(or at least I would, if I were the Referee)